#72259283

order	10816
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	D₄×C₁₃² ⋊ C₂×C₄
Sylow	C₁₃² ⋊ #442

Subgroups

	subgroup	G/H
center	C₂	#72214198
commutator	C₄×C₁₃²	C₂²×C₄
Frattini	C₄	C₂×D₁₃⋊#261
Fitting	D₄×C₁₃²	C₂×C₄
socle	C₂×C₁₃²	C₄×D₄
prime radical	#72259265	C₂
cosocle	C₄×C₁₃⋊D₁₃	C₂³
hypercenter	D₄	D₁₃⋊#261
hypocenter	C₁₃²	#442
LCS	G ▷ C₄×C₁₃² ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₄

546 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 104

Polycyclic presentation:

2^6,13^2|a^2=e,b^2=d,c^2=d,e^2=f,b^a=bef,c^a=cf,e^b=ef,g^a=g^12,g^b=g^8,g^c=g^8,g^d=g^12,h^a=g^6h,h^b=h^8,h^c=h^8,h^d=h^12

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259289, #72259559
maximal normal subgroup of	#72259289, #72259559

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 13, 13]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^12);;
gap> SetConjugate(coll, 7, 2, F.7^8);;
gap> SetConjugate(coll, 7, 3, F.7^8);;
gap> SetConjugate(coll, 7, 4, F.7^12);;
gap> SetConjugate(coll, 8, 1, F.7^6*F.8);;
gap> SetConjugate(coll, 8, 2, F.8^8);;
gap> SetConjugate(coll, 8, 3, F.8^8);;
gap> SetConjugate(coll, 8, 4, F.8^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);